To determine which equation represents a function, we need to identify which equation can be solved explicitly for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. Here is the step-by-step analysis of each option:
Option A: [tex]\( x = 13 \)[/tex]
- This equation gives a specific value to [tex]\( x \)[/tex] and does not express [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Option B: [tex]\( x - 12 = 34 \)[/tex]
- Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x - 12 = 34 \implies x = 46
\][/tex]
- Again, this constrains [tex]\( x \)[/tex] to a specific value and does not provide a relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex].
Option C: [tex]\( 2 y = -12 \)[/tex]
- Solving for [tex]\( y \)[/tex], we get:
[tex]\[
2y = -12 \implies y = -6
\][/tex]
- This equation shows that [tex]\( y \)[/tex] is a constant and does not depend on [tex]\( x \)[/tex]. It represents a horizontal line in a Cartesian coordinate system, which is indeed a function because for every [tex]\( y \)[/tex], there is a unique corresponding [tex]\( x \)[/tex] (in this case, [tex]\( x \)[/tex] does not affect [tex]\( y \)[/tex]).
Option D: [tex]\( 2 x - 4 x = 7 \)[/tex]
- Simplifying the equation, we get:
[tex]\[
2x - 4x = 7 \implies -2x = 7 \implies x = -\frac{7}{2}
\][/tex]
- This also provides a specific value for [tex]\( x \)[/tex] and does not establish a relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex].
Option E: [tex]\( \frac{x}{2} = 15 \)[/tex]
- Solving for [tex]\( x \)[/tex], we get:
[tex]\[
\frac{x}{2} = 15 \implies x = 30
\][/tex]
- Similarly, this defines [tex]\( x \)[/tex] as a constant and does not relate [tex]\( y \)[/tex] and [tex]\( x \)[/tex].
Only Option C, [tex]\( 2y = -12 \)[/tex], expresses a relationship where [tex]\( y \)[/tex] is a defined function, even if [tex]\( y \)[/tex] is a constant value. Therefore, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
